9. NORMAL SUBGROUPS AND FACTOR GROUPS
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Example. We use Theorem 8.3, its corollary, and Theorem 9.6 for the
following.
If m = n1n2 · · · nk where gcd(ni, nj ) = 1 for i 6= j, then
U (m) = Um/n1 (m)⇥Um/n2 (m)⇥· · ·⇥Um/nk (m) ⇡ U (n1) U (n2) · · · U (nk ).
We use the “=” sign for the internal direct product since the elements are all
within U (m).
U (105) = U (15) · U (7) = U15(105) ⇥ U7(105)
= {1, 16, 31, 46, 61, 76} ⇥ {1, 8, 22, 29, 43, 64, 71, 92}
⇡ U (7) U (15)
U (105) = U (5 · 21) = U5(105) ⇥ U21(105)
= {1, 11, 16, 26, 31, 41, 46, 61, 71, 76, 86, 101} ⇥ {1, 22, 43, 64}
⇡ U (21) U (5)
U (105) = U (3 · 5 · 7) = U35(105) ⇥ U21(105) ⇥ U15(105)
= {1, 71} ⇥ {1, 22, 43, 64} ⇥ {1, 16, 31, 46, 61, 76}
⇡ U (3) U (5) U (7)